# Do we need to check for all kinds of symmetry in a compound to declare it achiral?

I know that a compound is achiral if it contains any one of the following kind of symmetry.

1.Plane of symmetry

2.Centre of symmetry

3.Normal axis of symmetry

4.Alternate axis of symmetry

It is difficult to write, for many compounds, its mirror image and then check using visualisation that if the image and compound are same or not, so we check for these 4 symmetries in molecule. My question is that for any compound do we need to check for all the symmetry elements one by one till we get a symmetry possessed by the molecule to declare it achiral, for example, does there exist any compound which has only alternate axis of symmetry and not any other kind of symmetry, so that any one will have to check all the symmetries one by one, till he reaches alternate axis of symmetry, to declare it achiral?

I know compound which have centre of symmetry, but not plane of symmetry, but I don't know about a compound which has only alternate axis of symmetry and not any other kind of symmetry. Most of the achiral compounds that I have come across have plane of symmetry or centre of symmetry. So, do we need to check for all axis of symmetry or only centre and plane of symmetry, or any other shorter combination?

We need to check all 4 symmetries to determine whether a compound is optically active or not.

Consider the following compound. It has only Alternating Axis of Symmetry

Note that the rings are in perpendicular planes. The following diagrams illustrate the top view of the molecule.The stereochemistry of the methyl groups can be:

$$S_4$$ axis of symmetry is seen only in the following diastereomer where the methyl groups are anti to each other in each ring and the view looks like a Nazi cross from the top.

• Umm.. So? do we need to check or not.. you might have left that out.. also references please regarding the chirality of since it may may depend on the diastereomer taken.. – Safdar Faisal Sep 19 '20 at 12:28
• Your compound has undefined chiral centers; in which direction are the methyl pointing? – Raphaël Sep 19 '20 at 12:29
• I have edited the answer listing all possible cases where only one of the diastereomers has the S4 axis of symmetry – Soumyadwip Chanda Sep 19 '20 at 12:30
• Yes, that is a $C_2$ axis. But C axes are not taken under consideration while determining optical activity/chirality – Soumyadwip Chanda Sep 19 '20 at 12:48
• @SoumyadwipChanda A better choice of term would be 'swastik' rather than you know what :) – Aniruddha Deb Sep 19 '20 at 14:09